Problem 63 - Precalculus: A Concise Course

Question

\(g\) is related to a parent function \(f(x)=\sin (x)\) or \(f(x)=\cos (x)\) (a) Describe the sequence of transformations from \(f\) to \(g\). (b) Sketch the graph of \(g\). (c) Use function notation to write \(g\) in terms of \(f\). $$ g(x)=\cos (x-\pi)+2 $$

Step-by-Step Solution

Answer

The parent function for \(g(x)=\cos (x-\pi)+2\) is \(f(x)=\cos(x)\). \(g\) is obtained from \(f\) by a right-shift by \(\pi\) units and an upward-shift by 2 units. Thus, \(g(x) = f(x-\pi) + 2\).

Step 1: Identify the parent function and the transformation

Comparing \(g(x)=\cos (x-\pi)+2\) with \(f(x)=\cos(x)\), it is clear that the parent function \(f\) is \(f(x)=\cos(x)\). The function \(g\) is obtained from \(f\) by shifting it to the right by \(\pi\) units and upward by 2 units.

Step 2: Sketch the graph of \(g(x)\)

Starting with the graph of \(f(x)=\cos(x)\), shift it to the right by \(\pi\) units and upwards by 2 units. The new graph crosses the y-axis at \(y=1\) (because \(\cos(0)=1\) for the parent function) and it reaches maximum value at \(y=3\) and minimum at \(y=1\) instead of \(y=1\) and \(y=-1\) respectively for the parent function. Its period remains \(2\pi\).

Step 3: Write \(g\) in terms of \(f\)

Since \(g\) is obtained by right-shift and upward-shift from \(f\), we can write \(g\) in terms of \(f\) as follows: \(g(x) = f(x-\pi) + 2 = \cos(x - \pi) + 2\).

Key Concepts

Parent Function

Understanding the parent functions is crucial when studying the transformations of trigonometric functions. A parent function is the simplest form of a set of functions that form a family. For trigonometric functions, the most common parent functions are the sine function, \( f(x) = \sin(x) \), and the cosine function, \( f(x) = \cos(x) \). These basic forms have specific shapes and properties critical for identifying and predicting the outcomes of transformations. In our example, the parent function used is \( f(x) = \cos(x) \), which is a wave-like pattern that repeats every \( 2\pi \) units along the x-axis and has a range from -1 to 1.

Graphing Trigonometric Functions

Graphing trigonometric functions can seem daunting, but once you understand the parent function's graph, it becomes a matter of applying transformations. A sine or cosine function's graph shows the amplitude, period, phase shift, and vertical shift. By default, these functions oscillate between -1 and 1 and have a period of \( 2\pi \). When graphing, pay special attention to the points where the function crosses the x-axis (zeroes), reaches the maximum and minimum values (peaks and troughs), and where it intercepts the y-axis.

Function Notation

Function notation is a way to denote functions concisely and to facilitate the description of transformations. For example, \( g(x) \) is a transformation of the function \( f(x) \) in the exercise provided. The notation clearly represents how variable \( x \) is being manipulated within the function to produce specific outputs. Using function notation effectively allows us to connect the new function \( g \) with its parent function \( f \) and to describe precisely what changes have been applied.

Horizontal Shift

A horizontal shift occurs when a function is moved along the x-axis. In our given function \( g(x) = \cos(x - \pi) + 2 \), there's a horizontal shift to the right by \( \pi \) units. This process is also known as a phase shift in the context of trigonometric functions. Whether a function moves left or right depends on the sign of the horizontal shift; a positive value inside the function's argument translates to a left shift, while a negative value signifies a right shift.

Vertical Shift

A vertical shift refers to moving a function up or down along the y-axis. In our example, the \( +2 \) at the end of the function \( g(x) \) indicates that the parent function has moved upward by 2 units. This transformation doesn't affect the shape or horizontal position of the graph but alters the maximum and minimum values accordingly. The outcome is a new function that retains the features of the original one, but at a different vertical location on the Cartesian plane.